Turing predicted that a homogeneous and otherwise stable steady state of a reactive, kinetic system comprising an activator and inhibitor species may lose its stability and form inhomogeneous patterns due to the interaction of diffusion and reaction, see A. Turing, Philos. Trans. Roy. Soc. London Ser. B 237, 37 (1952). This prediction has been recently verified experimentally by Castets et al. wherein pattern formation was observed in the chlorite/iodide/malonic acid system, as disclosed in V. Castets, E. Dulos, J. Boissonade, and P. DeKepper, Phys. Rev. Lett. 64, 2953 (1990). These results support Turing's prediction that a homogeneous and otherwise stable reactive system may lose its stability (Turing instability) and form inhomogeneous patterns due to the interaction of diffusion and reaction.
This mechanism is believed to be fundamental to morphogenesis in biological systems as discussed in for example H. Meinhardt, Models of Biological Pattern Formation (Academic Press, 1982) and J. D. Murray, Mathematical Biology, (Springer Verlag, Berlin, 1989). L. Segel and J. L. Jackson, J. Theor. Biol. 37, 545 (1972). Specifically, the Turing instability can occur in an autocatalytic chemical or biochemical system comprising an activator species A which stimulates its own production and the production of its antagonist, an inhibitor I, if the diffusion coefficient of the inhibitor I is sufficiently greater than that of the activator A. When such a system is maintained in a homogeneous condition such as by stirring or other agitation means, it settles into a steady state through the balance of activation and inhibition. However, when a local concentration fluctuation arises in a non-agitated system, diffusion comes into play In addition to chemical relaxation. Diffusion acts to remove or replenish species that are locally in excess or depleted, respectively, as the case may be. When this process proceeds at different rates for the two species proportional to their diffusivities, the balance between the activator and inhibitor, that existed in the perfectly stirred system, may be broken in such a way as to permit the activator concentration to grow locally, see L. Segel and J. L. Jackson, J. Theor. Biol. 37, 545 (1972). Therefore, the primary role of diffusion in the instability is to spatially disengage the counteracting species.
The physical constraints under which the Turing instability is achieved are quite severe and can occur in an activator/inhibitor system only if the diffusion coefficient of the inhibitor is sufficiently larger than that of the activator, i.e. if .delta..ident.D.sub.inh /D.sub.act &gt;.delta..sub.c &gt;1. In other words, the Turing instability can only be observed in activator/inhibitor systems comprising different diffusivities of the activator and inhibitor species. However, in most liquid systems there is generally little difference in the diffusion coefficients for the different species as long as their molecular masses are of the same order, and the ratio D.sub.inh /D.sub.act is beyond experimental control: hence the Turing instability Is not a common occurrence.
Accordingly, it would be advantageous to provide a general method of destabilizing the homogeneous steady state or homogeneous oscillating states of spatially extended systems having kinetically or dynamically coupled components. It would be advantageous to provide such a method which would not depend primarily on the diffusion coefficients of the key species and which have general applicability to a wide range of systems such as chemical, physical and biological systems.